Question

the cubic polynomial f(x) is such that the coefficient of x cube is -1 and the roots of the equation f(x)=0 are 1, 2 & k.Given that f(x) has a remainder of 8 when divided by x-3, find the value of k.

Answer 1

The polynomial function is a cubic polynomial function with a general equation of y = ax^3 + bx^2 + cx + d. a is given to be -1. roots make y = 0

when x = 1
y = -1(1) + b + c + d = -1 +b+c+d = 0
when x = 2
y = -1*8 + 4b + 2c + d = -8 + 4b+c+d = 0
when x =k 
y = -k^3 + k^2b + kc + d = 0

when x = 3, y =8
8 = -27 + 9b + 3c + d

b=7/3 , c=23/3 , d=−9
k1 = 9.3556
k2 = -2.71051
k3 = 0.35491